{"id":"fa0abc66-9cb7-4bac-9631-d5608385023f","arxiv_id":"2412.01349","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A gauge-invariant catalogue of cosmological perturbations and stability conditions for teleparallel Horndeski gravity, with the gauge-fixed forms left partly to an external repository.","lead":"Gravitational perturbations around a uniform expanding universe are worked out in gauge-invariant form for BDLS teleparallel Horndeski gravity, a scalar-tensor alternative to Einstein's theory. The paper catalogues how scalar, vector, and tensor ripples behave and lists conditions for avoiding ghost and gradient instabilities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's Newtonian and synchronous gauge actions (App. B.1.c, B.1.d) yield three propagating scalar DoFs while the gauge-invariant section yields two; this unresolved internal inconsistency undermines the advertised gauge catalogue and the 9-DoF count.","rationale":"The paper presents a lengthy and carefully organized derivation of gauge-invariant cosmological perturbations for BDLS teleparallel Horndeski gravity. The gauge-invariant scalar action (54) and its reduced two-field form (69) are the core result, and the stability conditions (80)-(82), (89)-(91), (64) follow a standard high-k procedure. However, the paper's explicit purpose is to also provide a catalogue of gauges for observers, and that catalogue is where the central claim becomes fragile. The Newtonian and synchronous gauge results in Appendix B are not merely missing coefficients; they are inconsistent with the gauge-invariant sector in the number of propagating modes. The authors acknowledge the inconsistency but leave it unresolved, stating that further investigation is required. This is a load-bearing issue because if the extra mode is a residual gauge artifact, the catalogue as presented will mislead users into modeling a non-existent DoF; if it is physical, the gauge-invariant elimination in Sec. IV.A is incomplete and the 9-DoF count is wrong. I do not think the spin-connection assumption is the weakest point: fixing the Weitzenböck gauge is standard in teleparallel perturbation theory and can be justified by local Lorentz gauge fixing, though a dedicated check would still be welcome. The unresolved mode-count discrepancy is internal and concrete, and it is the main obstacle to accepting the advertised catalogue. The verdict remains CONDITIONAL as the reader said, pending a resolution of this inconsistency and the availability of the external coefficient repository.","tokens_in":39196,"tokens_out":7830,"duration_ms":69483,"concrete_test":"Independently recompute the Newtonian gauge scalar sector: impose E=0 and β=B on Eq. (52), vary with respect to Φ to get the constraint, substitute back, and compute the rank of the 3x3 kinetic matrix for {δφ, β, ψ} in the high-k limit. If the rank is 2, the third mode is pure gauge or auxiliary; if rank is 3, the Sec. IV.A elimination is incomplete. Also construct the residual gauge transformation preserving E=0 and β=B; if a nontrivial combination of ξ0 and ξ remains, that mode is a gauge artifact and the gauge is over-counted in App. B.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that BDLS theory has exactly 9 propagating DoFs (2 scalars, 1 pseudoscalar, 2 vector pairs, 2 tensors) and that the gauge-invariant quadratic actions and stability conditions are correct. This is intended to be supported by a catalogue of gauge choices in Appendix B. However, the catalogue is internally inconsistent with the gauge-invariant core. In Newtonian gauge (E=0, β=B), Eq. (B7), after eliminating the single auxiliary mode Φ, the authors obtain a final action (B8) with three dynamical scalar modes {δφ, β, ψ}. In synchronous gauge (Φ=0, β=B), Eq. (B9), they likewise obtain three dynamical scalars (B10). The gauge-invariant analysis in Sec. IV.A, after eliminating X2 and X4, yields exactly two propagating scalars X1 and X3 (Eqs. 69-73). The authors explicitly note the discrepancy in App. B and defer it to future work, saying it 'does not occur for widely used subclasses of BDLS.' This is not adequate: a valid gauge fixing cannot change the number of physical DoFs. The discrepancy means either the Newtonian/synchronous gauge fixing leaves a residual gauge freedom (making the third 'dynamical' mode pure gauge), or the gauge-invariant elimination of auxiliary fields misses a propagating mode. In the first case the advertised catalogue is misleading; in the second the central DoF count is wrong. The load-bearing result is therefore not robustly established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs the quadratic cosmological perturbations of the BDLS action, the teleparallel analogue of Horndeski gravity, on a flat FLRW background. It performs an SVT decomposition of tetrad perturbations, defines gauge-invariant variables for the scalar, vector, and tensor sectors, and derives gauge-invariant quadratic actions. From these actions the paper extracts propagating degrees of freedom and formulates ghost and Laplacian stability conditions in the high-k limit, concluding that the full BDLS theory carries 9 propagating degrees of freedom: 2 scalars, 1 pseudoscalar, 2 vector pairs, and 2 tensors. Appendix B provides a catalogue of common gauge choices (flat, unitary, Newtonian, synchronous, and vector gauges), with final coefficients partly stored in an external repository. The central unresolved issue is that the Newtonian and synchronous gauge actions in Appendix B contain three propagating scalar modes, while the gauge-invariant scalar analysis of Section IV gives two; this discrepancy is acknowledged but deferred to future work.","tokens_in":39451,"tokens_out":3017,"duration_ms":29177,"significance":"If correct, the paper would provide a useful reference for cosmological perturbations in teleparallel Horndeski gravity, complementing the earlier gauge-specific analysis of Ref. [4] and giving practitioners gauge-invariant results and stability conditions for a broad class of models. The explicit comparison with subclasses (GR, f(T), Horndeski, NGR) in Table I is a valuable summary. However, the advertised catalogue is undermined by the internal inconsistency between the gauge-invariant scalar sector and the Newtonian/synchronous gauge results, whose resolution is essential before the 9-DoF claim can be considered established. The paper also has the strength of connecting to published stability conditions for tensor and vector sectors and of providing an openly available repository for the lengthy coefficients, although the dependence on a non-archival repository is a weakness.","major_comments":[{"comment":"The Newtonian gauge action (B7)-(B8) and the synchronous gauge action (B9)-(B10) each yield three dynamical scalar modes after eliminating auxiliary fields, whereas the gauge-invariant analysis in Section IV.A yields exactly two propagating scalars, Ψ1 and Ψ2, after eliminating X2 and X4. The manuscript acknowledges this discrepancy but states only that it does not occur for widely used subclasses and defers resolution to future work. A gauge fixing cannot change the number of physical degrees of freedom, so the discrepancy must be resolved in the present manuscript: either the third mode is pure gauge due to residual gauge freedom (which must be demonstrated explicitly), or the gauge-invariant elimination of X2 and X4 misses a propagating mode (which would invalidate the central DoF count). Merely noting the issue and deferring it is not sufficient for a paper whose central claim is the 9-DoF count and whose Appendix B is advertised as a catalogue.","section":"Appendix B.1.c and B.1.d, Eqs. (B7)-(B10); compare Section IV.A, Eqs. (69)-(73)"},{"comment":"The perturbation calculation fixes the spin connection to the Weitzenböck gauge and perturbs only the tetrad components according to Eq. (49). The Conclusion explicitly limits the analysis to the Weitzenböck gauge for the spin connection. If the spin connection acquires perturbations that carry physical degrees of freedom, the derived actions and the 9-DoF count would be incomplete. This is a legitimate scope limitation, but it must be stated more prominently in the abstract and in the statement of the central claim, because the phrase \"full BDLS framework\" in the Conclusion overstates the validity of the result under this assumption.","section":"Section III and Conclusion, around Eq. (49)"},{"comment":"The final reduced actions in the flat, unitary, Newtonian, and synchronous gauges are stated with coefficients δ˜Fi, δ˜Ui, δ˜Ni, and δ˜Si all deferred to Ref. [37], an external GitHub repository. For a catalogue paper whose stated purpose is to give practitioners ready-to-use perturbed actions in different gauges, this is a major omission: the central results cannot be checked or used without downloading a repository that is not part of the archival record. The coefficients should be included in the manuscript or in an arXiv ancilliary file, or the verbal claims about the gauge catalogue must be substantially weakened.","section":"Appendix B, Eqs. (B3), (B6), (B8), (B10)"}],"minor_comments":[{"comment":"The title header contains a typo: \"g ravity\" should read \"gravity\", and the title itself should be checked for spacing.","section":"Title and abstract"},{"comment":"The phrase \"For several deacdes\" is a typo for \"For several decades\", and the prose would benefit from a careful proofreading pass.","section":"Introduction, first paragraph"},{"comment":"The word \"representes\" should be \"represents\".","section":"Section II.B, paragraph after Eq. (28)"},{"comment":"The action is written in position space with terms such as β/a^2 ∇^2(...), but later Fourier transforms are used; it would help to state explicitly the sign conventions for ∇^2 and the Fourier normalization early in the section.","section":"Section III.B, Eq. (52)"},{"comment":"The vector action uses both ∇v and (∇v)^2 where v likely denotes a vector magnitude; the notation is ambiguous and should define v, w, V as vectors, e.g., v = v_i, and clarify that ∇ denotes the spatial gradient.","section":"Section III.D, Eq. (58)"},{"comment":"There is a typo in the coefficient U8: \"GTeleTtvec\" should likely be \"GTele,Tvec\" or \"GTele,T Tvec\" depending on the intended derivative; this should be corrected.","section":"Appendix B.1.d, Eq. (B5h)"},{"comment":"The final coefficients of the gauge-fixed actions are only available via an external GitHub repository; even if the repository is retained, it would be preferable to include a stability or version identifier for the exact version used.","section":"Appendix B, general"}],"recommendation":"major_revision","confidential_remarks":"The core calculation is extensive and the paper fills a genuine gap. However, the unresolved discrepancy between the gauge-invariant scalar DoF count and the Newtonian/synchronous gauge counts is exactly the kind of issue that must be fixed before publication. I would also urge the editor to insist that the coefficients of the advertised gauge catalogue be included in the archival record rather than stored in a GitHub repository. The self-citation pattern is heavy but appropriate for a line of work by the same group; the main reservation is substantive, not stylistic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the gauge-invariant perturbation actions for BDLS teleparallel Horndeski are a genuine and usable addition to the literature, and the stability conditions are concrete enough to plug into model-building. But the advertised gauge catalogue is not finished. The Newtonian and synchronous sections produce three propagating scalars against the two from the gauge-invariant core, and the authors flag this and move on. With the final gauge-fixed coefficients also parked in an unversioned GitHub repo, the catalogue reads as a pointer rather than a deliverable.\n\nWhat is actually new: Ref [4] did the perturbations in one gauge; here you get the full gauge-invariant scalar and vector actions (Eqs. 54 and 60), the elimination of the auxiliary fields, and the ghost/gradient conditions (80-82, 89-91, 64). The coefficient lists in Appendix A are long but explicit, and the DoF table for subclasses GR, f(T), f(phi,X,T), NGR, and Horndeski is useful and consistent with the literature. The paper is also honest about its assumptions: the spin connection is held in the Weitzenbock gauge, and the diagonalized scalar action is only given in the high-k limit.\n\nThe soft spot, in proportion: the 3-vs-2 scalar discrepancy is load-bearing, because a valid gauge fixing cannot change the number of physical DoFs. Either the Newtonian/synchronous fixing leaves a residual gauge mode that only looks dynamical, or the gauge-invariant elimination drops a real mode. The gauge-invariant route is likely the right one, since nothing in the paper suggests the auxiliary elimination is wrong, which would make the gauge-fixed results misleading rather than the count wrong. But the paper does not resolve it, and saying it does not occur for widely used subclasses is not an answer for the general BDLS theory it claims to catalogue. This is a fixable problem, not a dead end: the likely resolution is residual gauge freedom in those two gauges, and the paper even notes synchronous residual freedom earlier in the appendix without connecting the dots.\n\nAlso worth saying: the coefficient content of the final gauge-fixed actions (B3, B6, B8, B10) is not in the paper. For a catalogue, that is a self-containment failure, and an unversioned repo makes it worse.\n\nWho it is for: anyone building or testing concrete BDLS models, and anyone needing the stability conditions. The gauge-invariant core deserves citation and use; the catalogue needs repair first. Send it to a referee, the calculation is substantial, the issues are identifiable, and a serious referee could push it to a trustworthy state.","headline":"The gauge-invariant perturbation actions for BDLS teleparallel Horndeski are a genuine new reference, but the gauge-fixed catalogue disagrees with its own scalar DoF count and the missing coefficients sit in an unversioned repo.","tokens_in":40057,"tokens_out":6748,"would_cite":true,"duration_ms":56300,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.50.Kd","98.80.-k"],"model":"deepseek-v4-flash","headline":"This paper derives the gauge-invariant cosmological perturbations of the teleparallel analogue of Horndeski gravity and finds nine propagating degrees of freedom in the general BDLS theory.","keywords":["teleparallel gravity","Horndeski gravity","cosmological perturbations","gauge invariance","ghost instabilities","degrees of freedom","BDLS theory"],"falsifier":"Recompute the second-order action with a general perturbed spin connection around the same FLRW tetrad and check whether any new kinetic terms appear; if they do, the nine-degree-of-freedom count changes. A more direct observational test would be the detection of a propagating vector or pseudoscalar polarisation in the cosmological gravitational-wave background, which the standard two-tensor GR prediction cannot produce.","tokens_in":38971,"feed_emoji":"🌌","tokens_out":7467,"duration_ms":67255,"temperature":0.7,"pith_summary":"This paper works out, in gauge-invariant form, the cosmological perturbations of the BDLS action, the teleparallel analogue of Horndeski gravity. It claims that the most general BDLS theory carries nine propagating degrees of freedom around a flat Friedmann-Lemaitre-Robertson-Walker background: two scalar modes, one pseudoscalar mode, two pairs of vector modes, and the two tensor polarisations. It also derives the quadratic actions for each sector and the coefficient inequalities that prevent ghost and gradient instabilities. The result matters because it converts the BDLS framework from a background-only theory into one whose perturbations can be compared with cosmological observations and implemented in numerical codes.","feed_headline":"Teleparallel Horndeski cosmology carries nine propagating modes","feed_subtitle":"Full gauge-invariant perturbation catalogue yields scalars, vectors, pseudoscalar and tensor modes with stability conditions.","key_machinery":"The central object is the tetrad perturbation decomposition of Eq. (49), which splits the sixteen tetrad components into five scalars ($\\Phi$, $\\psi$, $B$, $\\beta$, $E$), one pseudoscalar $\\sigma$, three vectors ($u_i$, $v_i$, $w_i$), one pseudovector $V_i$, and the tensor $h_{ij}$, with the background tetrad fixed in the Weitzenböck gauge. Gauge-invariant combinations $X_1,\\dots,X_4$ for scalars and $Y_i$, $Z_i$ for vectors remove coordinate freedom; then the auxiliary fields $X_2$, $X_4$, and $Y_i$ are integrated out, and the kinetic matrix is diagonalised at high $k$ so that the propagating modes and their sound speeds can be read off from the resulting coefficients.","core_discovery":"On the paper's own terms, the central discovery is that the BDLS action, despite its many coupling functions, has a tractable scalar-vector-tensor decomposition: after eliminating auxiliary fields in the high-k limit, the scalar sector leaves two propagating scalars, the pseudoscalar sector leaves one mode, the vector sector leaves two pairs of modes, and the tensor sector leaves the two usual polarisations. The corresponding gauge-invariant quadratic actions are Eqs. (54), (60), and (61), and the ghost and gradient stability requirements reduce to Eqs. (80)-(82) for scalars, Eqs. (89)-(91) for vectors, Eq. (64) for tensors, and $B_1 > 0$ for the pseudoscalar. The paper also confirms that subclasses reduce the count: GR and $f(T)$ give two degrees of freedom, $f(\\varphi,X,T)$ and Horndeski give three, generalized teleparallel dark energy and generalized scalar-tensor theory give three, and New General Relativity gives eight.","pith_inferences":["Beyond the paper: because the background tetrad is fixed in the Weitzenböck gauge, a full calculation that also perturbs the spin connection is the direct next test; new kinetic terms would change the nine-mode count.","Beyond the paper: the gauge catalogue is detailed enough to be coded directly into a Boltzmann solver, and comparing the resulting CMB and matter power spectra with the standard cosmological model would quantify how strongly the extra scalar and vector modes are suppressed.","Beyond the paper: the same elimination procedure can be applied at higher order in perturbations, opening the way to bispectra in BDLS; the present second-order action is the necessary first step."],"forward_implications":["The full BDLS theory has nine propagating degrees of freedom: two scalars, one pseudoscalar, two vector pairs, and two tensor polarisations; known subclasses reduce this count, from two for GR and $f(T)$ up to eight for New General Relativity.","Ghost and gradient stability become concrete coefficient checks: the scalar sector is stable when Eqs. (80)-(82) hold, the vector sector when Eqs. (89)-(91) hold, the tensor sector when Eq. (64) holds, and the pseudoscalar requires $B_1 > 0$.","The tensor sound speed $c_T^2 = D_2/D_1$ is generically not unity and depends on the teleparallel couplings, so gravitational-wave observations can directly constrain the BDLS Lagrangian.","The gauge catalogue in Appendix B provides flat, unitary, Newtonian, and synchronous versions of the same perturbation equations, allowing the same physics to be implemented in different observer frames.","Observational codes can now use these actions to compute power spectra and stability conditions for specific BDLS models, which is the step needed to confront the framework with structure-formation data."],"supporting_citations":[{"why":"introduces the BDLS action whose perturbations are the subject of this paper","marker":"[18]"},{"why":"previous computation of cosmological perturbations in a fixed gauge, which this work extends to gauge-invariant form","marker":"[4]"},{"why":"derives the gravitational-wave speed constraint in teleparallel Horndeski, supplying the tensor-sector comparison","marker":"[15]"},{"why":"computes dispersion relations and polarisations, giving the degree-of-freedom count that this paper confirms","marker":"[13]"},{"why":"analyses stability and degrees of freedom in New General Relativity, the NGR subclass comparison","marker":"[10]"},{"why":"f(T) perturbation analysis used for the f(T) degree-of-freedom and longitudinal-gauge comparison","marker":"[61]"},{"why":"standard Horndeski quadratic action recovered in the $G_{\\mathrm{Tele}} \\to 0$ limit, anchoring the scalar and tensor coefficient checks","marker":"[64]"}],"fun_headline_variants":["Teleparallel Horndeski gravity yields nine propagating modes","Nine modes in teleparallel Horndeski: full gauge-invariant catalogue","BDLS gravity's nine modes: scalars, vectors, tensors and a pseudoscalar","Gauge-invariant perturbations in teleparallel Horndeski: nine modes","Nine propagating modes in teleparallel Horndeski perturbations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole calculation keeps the spin connection fixed in the Weitzenböck gauge and perturbs only the tetrad; if the spin connection carries its own physical perturbations, the mode count and stability conditions could be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Teleparallel Horndeski gravity yields nine propagating modes","Nine modes in teleparallel Horndeski: full gauge-invariant catalogue","BDLS gravity's nine modes: scalars, vectors, tensors and a pseudoscalar","Gauge-invariant perturbations in teleparallel Horndeski: nine modes","Nine propagating modes in teleparallel Horndeski perturbations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000698,"raw_usage":{"total_tokens":3116,"prompt_tokens":871,"completion_tokens":2245,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":2147}},"tokens_in":487,"tokens_out":2245,"duration_ms":13149,"temperature":1.0,"reasoning_tokens":2147,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:26:39.484979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the second-order action with a general perturbed spin connection around the same FLRW tetrad and check whether any new kinetic terms appear; if they do, the nine-degree-of-freedom count changes. A more direct observational test would be the detection of a propagating vector or pseudoscalar polarisation in the cosmological gravitational-wave background, which the standard two-tensor GR prediction cannot produce.","supporting_citations":[{"cited_title":"Dialektopoulos , Celia Escamilla-Rivera, Gabriel Farrugia, Viktor Gakis, Martin Hendry, Manuel Hohmann, Jackson Levi Said, Jurgen Mifsud, a nd Eleonora Di Valentino","cited_arxiv_id":null,"evidence_quote":"introduces the BDLS action whose perturbations are the subject of this paper"},{"cited_title":"Dialektopoulos, Ja ckson Levi Said, Abdurakhmon Nosirov, Zinovia Oikonomopoulou, and Odil Yunusov","cited_arxiv_id":null,"evidence_quote":"previous computation of cosmological perturbations in a fixed gauge, which this work extends to gauge-invariant form"},{"cited_title":"Dialektopoulos , Viktor Gakis, and Jackson Levi Said","cited_arxiv_id":null,"evidence_quote":"derives the gravitational-wave speed constraint in teleparallel Horndeski, supplying the tensor-sector comparison"},{"cited_title":"Alves Batista et al","cited_arxiv_id":null,"evidence_quote":"computes dispersion relations and polarisations, giving the degree-of-freedom count that this paper confirms"},{"cited_title":"New general relat ivity","cited_arxiv_id":null,"evidence_quote":"f(T) perturbation analysis used for the f(T) degree-of-freedom and longitudinal-gauge comparison"},{"cited_title":"Generalized G-inflation: Inflation with the most general second-order field equations","cited_arxiv_id":null,"evidence_quote":"standard Horndeski quadratic action recovered in the $G_{\\mathrm{Tele}} \\to 0$ limit, anchoring the scalar and tensor coefficient checks"}],"review_version":1}